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296 11. COUNTING AND CLASSIFICATIONR code11.28exp( coef(m10.8)[1] )a34.8And by no small coincidence, this is the same value you get by just empirically averaging theTotal.Tools column.Adding a predictor is as you’d expect by now, just add terms to the second line. Forexample, here is the second model we’d like to fit, using log-population as a predictor:And the code to fit it:n i ∼ Poisson(λ i )log λ i = α + β P log P iR code11.29m10.9
11.2. POISSON 297compare(m10.8,m10.9,m10.10,m10.11,nobs=nrow(d) , DIC=TRUE )R code11.31k AICc w.AICc dAICc DIC pD wDIC dDICm10.10 3 81.05 0.76 0.00 76.99 2.97 0.53 0.00m10.9 2 83.77 0.20 2.72 82.07 2.00 0.04 5.08m10.11 4 86.91 0.04 5.86 77.44 3.26 0.42 0.45m10.8 1 133.92 0.00 52.87 133.32 0.95 0.00 56.33ese rankings suggest that at least log-population is a useful predictor, and possibly contactrate as well.First, you can inspect the estimates and see if the coefficients are in the predicted directions.Comparing estimates across models:coeftab(m10.8,m10.9,m10.10,m10.11)R code11.32m10.8 m10.9 m10.10 m10.11a 3.55 1.33 0.93 0.94bp NA 0.24 0.27 0.26bc NA NA 0.29 -0.09bcp NA NA NA 0.04nobs 10 10 10 10And since it’s hard to see the uncertainty around these estimates, FIGURE 11.8 shows theresult of plotting the coeftab. In models m10.9 and m10.10, the main effects of both logpopulationand contact rate are consistent with the hypotheses: they are both positive, beingassociated with more tools. Interpreting m10.11 is harder, because we didn’t center the predictorsbefore fitting (remember centering, from Chapter 4). But the interaction estimate, atleast, can be interpreted. It is positive, as it should be if high contact rate and population sizeact synergistically. But the size of the estimate is very small, and if you inspect the standarderror on it, you’ll see it’s very uncertain.To get a sense for what’s being predicted, let’s plot the model predictions, as usual. ereare no new coding tricks here, aside from the detail of how to convert the linear model intothe prediction scale. To illustrate that, I’m going to produce two plots. In the first, I’ll drawthe mean and 95% confidence interval of the mean, using the usual lines. In the second, I’llplot Poisson random samples from the posterior, so you can see how to simulate observationsfrom a fit Poisson model.To calculate predicted mean counts, you just need the usual sapply code, now with theinverse link function embedded. is is exactly what you did when you plotted the binomialmodels. But now the link function isn’t the logit, but rather the logarithm. So this code willcompute the predicted mean and 95% confidence interval of the mean, for low contact andhigh contact islands, respectively:post
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Statistical RethinkingA BAYESIAN CO
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4 CONTENTS5.5. Ordinary least squar
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PrefaceMasons, when they start upon
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HOW TO USE THIS BOOK 9least a minor
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HOW TO USE THIS BOOK 11than initial
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1 e Golem of PragueIn the 16th cent
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16 1. THE GOLEM OF PRAGUEconverses
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1.2. WRECKING PRAGUE 19model P 1B ,
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1.2. WRECKING PRAGUE 21e dominant r
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1.3. THREE TOOLS FOR GOLEM ENGINEER
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2 Small Worlds and Large WorldsWhen
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2.1. PROBABILITY IS JUST COUNTING 3
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2.2. COLOMBO’S FIRST BAYESIAN MOD
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W L W W W L W L W2.2. COLOMBO’S F
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2.3. COMPONENTS OF THE MODEL 41not
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2.3. COMPONENTS OF THE MODEL 43You
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2.3. COMPONENTS OF THE MODEL 45e ma
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2.4. MAKING THE MODEL GO 47other pr
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2.4. MAKING THE MODEL GO 495 points
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2.4. MAKING THE MODEL GO 51just the
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2.4. MAKING THE MODEL GO 53between
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2.6. PRACTICE 55Medium.2.6.5. m1. R
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2.6. PRACTICE 57implied by the equa
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3 Sampling the ImaginaryLots of boo
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3. SAMPLING THE IMAGINARY 61a probl
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3.2. SAMPLING TO SUMMARIZE 63plot(
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3.2. SAMPLING TO SUMMARIZE 65Densit
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3.2. SAMPLING TO SUMMARIZE 67In con
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3.2. SAMPLING TO SUMMARIZE 69Densit
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3.3. SAMPLING TO SIMULATE PREDICTIO
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3.5. PRACTICE 793.5. PracticeEasy.
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3.5. PRACTICE 813.5.17. Predict sec
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84 4. LINEAR MODELSFIGURE 4.1. e Pt
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86 4. LINEAR MODELSexperiment with
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88 4. LINEAR MODELS4.1.4.2. Epistem
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90 4. LINEAR MODELSe approach above
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92 4. LINEAR MODELS'data.frame': 54
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94 4. LINEAR MODELSe point isn’t
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96 4. LINEAR MODELShave the samples
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98 4. LINEAR MODELSere’s no troub
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100 4. LINEAR MODELS)Note the comma
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102 4. LINEAR MODELSpreviously obse
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104 4. LINEAR MODELS4.4. Adding a p
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106 4. LINEAR MODELS(1) What is the
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108 4. LINEAR MODELSdata(Howell1)d
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110 4. LINEAR MODELSRethinking: Wha
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112 4. LINEAR MODELSheight140 150 1
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116 4. LINEAR MODELSYou end up with
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118 4. LINEAR MODELS(1) Use link to
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122 4. LINEAR MODELSRethinking: Lin
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126 4. LINEAR MODELS4.7.1. e1. In t
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5 Multivariate Linear ModelsOne of
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5.1. SPURIOUS ASSOCIATION 131Divorc
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5.1. SPURIOUS ASSOCIATION 133But me
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5.1. SPURIOUS ASSOCIATION 135abmbas
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5.1. SPURIOUS ASSOCIATION 137this i
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5.1. SPURIOUS ASSOCIATION 139slower
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5.1. SPURIOUS ASSOCIATION 141Median
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5.1. SPURIOUS ASSOCIATION 143(a)(b)
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5.2. MASKED RELATIONSHIP 145N
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5.2. MASKED RELATIONSHIP 147a ~ dno
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5.2. MASKED RELATIONSHIP 149dcc$log
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5.3. WHEN ADDING VARIABLES HURTS 15
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5.4. CATEGORICAL VARIABLES 161$ wei
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5.4. CATEGORICAL VARIABLES 1635.4.2
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5.4. CATEGORICAL VARIABLES 165# sam
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168 5. MULTIVARIATE LINEAR MODELS5.
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170 5. MULTIVARIATE LINEAR MODELS5.
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6 Model Selection, Comparison, and
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6.1. THE PROBLEM WITH PARAMETERS 17
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6.2. INFORMATION THEORY AND MODEL P
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6.3. AKAIKE INFORMATION CRITERION 1
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6.4. DEVIANCE INFORMATION CRITERION
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6.5. USING AIC 203helps guard again
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6.5. USING AIC 205compare( m6.11 ,
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6.5. USING AIC 207e attitude this b
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6.5. USING AIC 209kcal.per.g0.5 0.7
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6.7. PRACTICE 211Consider by analog
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6.7. PRACTICE 2136.7.3. e deviance
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216 7. INTERACTIONSFIGURE 7.1. TOP:
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218 7. INTERACTIONSlog(rgdppc_2000)
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220 7. INTERACTIONSird, we may want
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222 7. INTERACTIONSlog GDP year 200
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226 7. INTERACTIONSInteraction mode
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228 7. INTERACTIONSthis model and t
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232 7. INTERACTIONSe main effect li
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234 7. INTERACTIONSbs -38.91 34.94s
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236 7. INTERACTIONSe primary reason
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238 7. INTERACTIONSNow for the plot
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240 7. INTERACTIONS7.4. Higher-orde
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8 Markov Chain Monte Carlo Estimati
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346 13. MULTILEVEL MODELSCompute th
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348 13. MULTILEVEL MODELSabsolute e
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350 13. MULTILEVEL MODELS)sigma_act
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14 Multilevel Models II: Slopes14.1
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14.1. EVERYTHING CAN VARY AND PROBA
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14.1. EVERYTHING CAN VARY AND PROBA
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360 15. MISSING DATA AND OTHER OPPO
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16 Writing Statistics379
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382 ENDNOTES12. For an autopsy of t
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384 ENDNOTES45. Fisher (1925), in C
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386 ENDNOTES77. See two famous edit
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388 ENDNOTES113. Hurlbert (1984) is
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390 BibliographyFrank, S. A. (2011)
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392 BibliographyProulx, S. R. and A
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IndexAkaike (1973), 380, 383Akaike
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INDEX 397non-identifiability, 153Oc
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